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Only weakly related, but good deck-building software (including the one I usually used when I still played MtG a lot, at Essential Magic) will give you sample opening hands and some statistical information.
–
AesinJul 12 '12 at 19:57

take the odds that you won't get any birds of paradise, and invert it
–
Sam I amApr 9 '13 at 21:39

3 Answers
3

The odds of drawing a particular card in a 60-card deck are obviously 1/60. If there are four such cards, the odds are 4/60. The odds of NOT drawing one of those cards in the first draw is 1 - 4/60 = 56/60.

To calculate the odds of the entire first hand, we can do it backwards:

The odds of not having any of the four cards in the first card is 56/60 (as I said above). The second card has odds of 55/59 (i.e. one of the remaining non-Bird cards after a non-Bird card was drawn to start), and then 54/58 and so on:

Card 1: 56/60 chance of not being the card you targeted

Card 2: 55/59

Card 3: 54/58

Card 4: 53/57

Card 5: 52/56

Card 6: 51/55

Card 7: 50/54

The odds of ALL of these happening (i.e. none of the four cards being in your hand) is the result of multiplying all these odds together:

(56*55*54*53*52*51*50) / (60*59*58*57*56*55*54) = ~0.6005 or ~60%

To calculate the odds of at least one of these cards being the one you're looking for, you can subtract this result from 1 (or 100%) to get a ~40% chance that (at least) one of your four cards will occur in a 7-card draw from a 60-card deck.

Magic Workstation besides many other tools for collection management, deck building, and online play has a very powerful probability calculator. It will go beyond opening hand and will let you see by what turn are you likely to have drawn the combo that you need.

The calculation you are looking for is called a Hypergeometric Distribution. This calculated your chances of drawing a particular number of "successes" from a population, without replacement.

Population Size: 60 cards

Successes in Population: 4 Birds of Paradise

Sample Size: 7 cards

Successes in Sample: exactly 1

Results: 33%

The online calculator will also give you the odds of drawing greater than (6%) the exact number of successes in the sample, and "at least" as many successes (greater than or equal to 1 = 40%).

You can see the calculation on the Wikipedia page, or searching math.stackexchange.com for Hypergeometric Distribution. Unfortunately, this site doesn't support math formatting. (Note: You will also need to know how to calculate binomial coefficients (and factorials).